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/ department of mathematics and computer science

Process Algebra (2IMF10)

algebra of natural numbers Bas Luttik

MF 6.072 s.p.luttik@tue.nl http://www.win.tue.nl/~luttik

Lecture 1

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Algebra of natural numbers

Consider set of natural numbers N together with the familiar operations

▶ + : N × N → N; and ▶ × : N × N → N.

Let succ : N → N be the unary operation on N associating with every natural number its successor (i.e., succ(n) = n + 1 for all n ∈ N). Then +, ×, succ and 0 satisfy the following equations for all m, n ∈ N: n + 0 = n m + succ(n) = succ(m + n) n × 0 = 0 m × succ(n) = (m × n) + m Since every n ∈ N is obtained by the n-fold application of succ to 0, this can be considered as a recursive defjnition of + and ×.

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Syntax

Defjnitions

A signature is a collection of symbols with an arity. A signature Σ determines, together with a countably infjnite set of variables V , an inductively defjned set T (Σ, V ) of terms (see book). The signature Σ1 of the theory natural numbers consists of

▶ a nullary function symbol (a.k.a. constant symbol) 0; ▶ a unary function symbol s; and ▶ binary function symbols a and m. 4/11 / department of mathematics and computer science

Equational theory

Defjnitions

An equation is a formula of the form t = u, with t and u terms. An equational theory is a pair (Σ, E) consisting of a signature Σ and a set of Σ-equations E; the equations in E are the axioms of the equational theory. The equational theory (Σ1, E 1) of natural numbers has the following axioms: a(x, 0) = x PA1 a(x, s(y)) = s(a(x, y)) PA2 m(x, 0) = 0 PA3 m(x, s(y)) = a(m(x, y), x) PA4

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Equational logic

Let T = (Σ, E) be an equational theory. We write T ⊢ t = u if there is a derivation of t = u using the following rules of equational logic: t = u (Axiom) if t = u is an equation in E t = t (Refm.) t = u u = t (Symm.) t = u u = v t = v (Trans.) t = u t[σ] = u[σ] (Subst.) σ : V → T (Σ, V ) is a substitution t1 = u1 · · · tn = un f (t1, . . . , tn) = f (u1, . . . , un) (Cont.) f ∈ Σ of arity n

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Equational logic (example derivation)

Consider the equational theory T1 = (Σ1, E 1). Give a proof tree showing that T1 ⊢ m(s(0), s(0)) = a(0, s(0)).

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Model: linking logic and algebra

We defjne an interpretation ι of the function symbols in Σ1 as functions

• n natural numbers:

0 → 0 , s → succ , a → + , and m → × . Then, under an assignment α : V → N, every term in T (Σ1, V ) denotes an element of N: ια(x) = α(x) ια(f (t1, . . . , tn)) = ι(f )(ια(t1), . . . , ια(tn))

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The quality of an equational theory

Let T = (Σ, E) be an equational theory, and assume some fjxed interpretation of the function symbols in Σ in a Σ-algebra1A. A Σ-equation t = u is valid in A (notation: A | = t = u) if ια(t) = ια(u) for all assignments α : V → N.

▶ T is sound for A: all equations that are derivable in T are valid in A. ▶ T is complete for A: all equations valid in A are derivable in T. ▶ T is ground-complete for A: all closed equations valid in A are

derivable in T.

1 A Σ-algebra is an algebra together with an interpretation of each n-ary

function symbol in Σ as an n-ary operation on A.

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T1 is sound for N

Let N = (N, +, ×, succ, 0), and let ι be the standard interpretation of a, m, s and 0 as +, ×, succ and 0 in N.

Theorem

For all Σ1-terms t and u, if T1 ⊢ t = u, then N | = t = u.

Proof.

By the general soundness of the rules of equational logic (see Proposition 2.3.9 in the book) it suffjce to check that the axioms PA1–PA4 are valid in N (Exercise!).

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T1 is ground-complete for N

Theorem

For all closed Σ1-terms p and q, if N | = p = q, then T1 ⊢ p = q. [Read the proof in the book. We will study a ground-completeness proof in the context of process algebra.]

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Completeness

Is T1 complete for N? No! For instance, N | = a(x, y) = a(y, x), but T1 ̸⊢ a(x, y) = a(y, x).

Exercise

Argue that T1 ̸⊢ a(x, y) = a(y, x).